Week 9 - Tree-Based Methods

Dr. David Elliott

1.1. Introduction

1.2. General Decision Tree Algorithm

1.3. Specific Decision Tree Algorithms

1.1. Introduction

Tree-based methods stratify or segment the predictor space into a number of simple regions¹.

As the spliting rules to make these decision regions can be summerised in a tree structure, these approaches are called decision trees.

A decision tree can be thought of as breaking data down by asking a series of questions in order to categorise samples into the same class.

NOTES

"In the context of the different categories of machine learning algorithms that we defined at the beginning of this course, we may categorize decision trees as follows:

• Supervised learning algorithm: classification and regression
• Optimization method: combinatorial -> greedy search
• Eager learning algorithm
• Batch learning algorithm
• Nonparametric model
• Deterministic (vs. stochastic)"

• Supervised learning algorithm:
• greedy search:
• Eager learning algorithm:
• Batch learning algorithm:
• Nonparametric model:
• Deterministic:

TODO

• in the notes explain what these are again.
• have a look through this: https://github.com/JWarmenhoven/ISLR-python/blob/master/Notebooks/Chapter%208.ipynb

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https://github.com/rasbt/stat479-machine-learning-fs19/blob/master/06_trees/06-trees__notes.pdf

Already Cloned

Terminology⁵

Root node: no incoming edge, zero, or more outgoing edges.

Internal node: one incoming edge, two (or more) outgoing edges.

Leaf node: each leaf node is assigned a class label if nodes are pure; otherwise, the class label is determined by majority vote.

Parent and child nodes: If a node is split, we refer to that given node as the parent node, and the resulting nodes are called child nodes.

Notes

• Leaves are typically drawn upside down, so they are at the bottom of the tree

Figure 1: Categorical Decision Tree

Dataset Example: Penguins

The "palmer penguins" dataset² contains data for 344 penguins from 3 different species and from 3 islands in the Palmer Archipelago, Antarctica.

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Artwork by @allison_horst

Figure 2: Penguin Buddies

	species	island	bill_length_mm	bill_depth_mm	flipper_length_mm	body_mass_g	sex
0	Adelie	Torgersen	39.1	18.7	181.0	3750.0	Male
1	Adelie	Torgersen	39.5	17.4	186.0	3800.0	Female
2	Adelie	Torgersen	40.3	18.0	195.0	3250.0	Female
3	Adelie	Torgersen	NaN	NaN	NaN	NaN	NaN
4	Adelie	Torgersen	36.7	19.3	193.0	3450.0	Female

1.3. General Decision Tree Algorithm

An algorithm starts at a tree root and then splits the data based on the features that give the best split based on a splitting criterion.

Generally this splitting procedure occours until^3,4...

• ...all the samples within a given node all belong to the same class,
• ...the maximum depth of the tree is reached,
• ...a split cannot be found that improves the model.

NOTES

• The process of growing a decision tree can be expressed as a recursive algorithm as follows⁵:
  1. Pick a feature such that when parent node is split, it results in the largest information gain and stopping if information gain is not positive.
  2. Stop if child nodes are pure or no improvement in class purity can be made.
  3. Go back to step 1 for each of the two child nodes.

TODO

• I made way to many specific functions below so refactor to make it a bit more managable!

Below is an example of a very shallow decision tree where we have set max_depth = 1.

Terminology (Reminder)¹

• The regions $R_1$ and $R_2$ are known as terminal nodes or leaves of the tree.
• The points where the predictor space is split are known as the internal nodes. In this case as we only have one split this is the "root node".
• The segments of the trees that connect the nodes are branches.

Extra: dtreeviz

Heres an additional visualisation package with extra features such as bein able to follow the path of a hypothetical test sample.

I don't use dtreeviz in the lectures, as it can be a bit of a hassle to setup. However you may also find this a useful way of thinking about the splitting.

Notes

• "For classifiers, however, the target is a category, rather than a number, so we chose to illustrate feature-target space using histograms as an indicator of feature space distributions." https://explained.ai/decision-tree-viz/index.html

We can make the tree "deeper", and therefore more complex, by setting the max_depth = 3.

We could also use more than 2 features as seen below.

NOTES

• Although more features are harder to plot decision boundaries (see dtreeviz).

We could also also easily extend this to have more than a 2 (binary) class labels.

In a general sense this approach is pretty simple, however there are a number of design choices and considerations we have to make including⁵:

• How do we decide which features to select for splitting a parent node into child nodes?
• How do we decide where to split features?
• When do we stop growing a tree?
• How do we make predictions if no attributes exist to perfectly separate non-pure nodes further?

1.4. Specific Decision Tree Algorithms

Most decision tree algorithms address the following implimentation choices differently⁵:

• Splitting Criterion: Information gain, statistical tests, objective function, etc.
• Number of Splits: Binary or multi-way.
• Variables: Discrete vs. Continuous.
• Pruning: Pre- vs. Post-pruning.

There are a number of decision tree algorithms, prominant ones include:

• Iterative Dichotomizer 3 (ID3)
• C4.5
• CART

Notes

• Binary means nodes always have two children.

CART

Scikit-Learn uses an optimised version of the Classification And Regression Tree (CART) algorithm.

• Splitting Criterion: Information gain
• Number of Splits: Binary
• Independent Variables (Features): Continuous
• Dependent variable: Continuous or Categorical
• Pruning: Pre- & Post-pruning

Notes

• "scikit-learn uses an optimised version of the CART algorithm; however, scikit-learn implementation does not support categorical variables for now." https://scikit-learn.org/stable/modules/tree.html

Information Gain⁴

An algorithm starts at a tree root and then splits the data based on the feature, $f$, that gives the largest information gain, $IG$.

To split using information gain relies on calculating the difference between an impurity measure of a parent node, $D_p$, and the impurities of its child nodes, $D_j$; information gain being high when the sum of the impurity of the child nodes is low.

We can maximise the information gain at each split using,

$$IG(D_p,f) = I(D_p)-\sum^m_{j=1}\frac{N_j}{N_p}I(D_j),$$

where $I$ is out impurity measure, $N_p$ is the total number of samples at the parent node, and $N_j$ is the number of samples in the $j$th child node.

Some algorithms, such as Scikit-learn's implimentation of CART, reduce the potential search space by implimenting binary trees:

$$IG(D_p,f) = I(D_p) - \frac{N_\text{left}}{N_p}I(D_\text{left})-\frac{N_\text{right}}{N_p}I(D_\text{right}).$$

Three impurity measures that are commonly used in binary decision trees are the classification error ($I_E$), gini impurity ($I_G$), and entropy ($I_H$)⁴.

Classification Error⁴

This is simply the fraction of the training observations in a region that does belongs to the most common class:

$$I_E = 1 - \max\left\{p(i|t)\right\}$$

Here $p(i|t)$ is the proportion of the samples that belong to the $i$th class $c$, for node $t$.

Notes

• Another way of writing this is $E = 1 - \substack{max\\k}(\hat p_{mk})$, where $\hat p_{mk}$ is the proportion of training observations in the $m$th region that are from the $k$th class¹.

Entropy Impurity⁴

For all non-empty classes ($p(i|t) \neq 0$), entropy is given by

$$I_H=-\sum^c_{i=1}p(i|t)\log_2p(i|t).$$

The entropy is therefore 0 if all samples at the node belong to the same class and maximal if we have a uniform class distribution.

For example in binary classification ($c=2$):

• $I_H=0 \text{ if } p(i=1|t)=1 \text{ or } p(i=0|t)=0$
• $I_H=1 \text{ if } p(i=1|t)=0.5 \text{ or } p(i=0|t)=0.5$

Notes

• In binary classification, entropy reaches its minimum (0) when all examples in a given node have the same label; on the other hand, the entropy is at its maximum of 1 when exactly one-half of examples a node are labeled with 1which would be useless for classification.⁶
• Another way of writing this is¹ $D = -\sum^K_{k=1}\hat p_{mk}log \hat p_{mk}$.
  • Entropy will take on a value near 0 if the $\hat p_{mk}$'s are all near 0 or 1, therefore will take on a small value if the $m$th node is pure.

Gini Impurity⁴

Gini Impurity is an alternative measurement, which minimises the probabilty of misclassification,

$$ \begin{align} I_G(t) &= \sum^c_{i=1}p(i|t)(1-p(i|t)) \\ &= 1-\sum^c_{i=1}p(i|t)^2. \end{align} $$

This measure is also maximal when classes are perfectly mixed (e.g. $c=2$):

$$ \begin{align} I_G(t) &= 1 - \sum^c_{i=1}0.5^2 = 0.5. \end{align} $$

Notes

• it is also a measure of node "purity" as a small value indicates a node contains predominantly observations from a single class.
• Another way of writing this is¹ for $K$ classes, $G = \sum^K_{k=1}\hat p_{mk}(1-\hat p_{mk})$.
• It takes a small value if all of the $\hat p_{mk}$'s are close to 0 or 1.
• Whether we use entropy or Gini impurity does not really matter, because both have the same concave/bell shape.
• Gini is computationally more efficient to compute than entropy due to the lack of the log.

Why not Classification Error?

Classification Error is rarely used for information gain in practice.

This is because it can mean that tree growth gets stuck and error doesnt improve, this is not the case for a concave function such as entropy or gini.

Notes

• Classification Error its not even an option in scikit-learn
• Another example of this is given in exersise x

Figure 12: Child Node Averages

Feature Importance

Decision trees allow us assess the importance of each feature for classifying the data.

The importance of a feature is the normalized total reduction of the criterion (e.g. Gini) brought by that feature¹⁰.

[INSERT ALGEBRA]

bill_length_mm: 0.015
bill_depth_mm: 0.030
flipper_length_mm: 0.955
body_mass_g: 0.000
total: 1.0

Pruning

Question: When do we stop growing a tree?

Occam’s razor: Favor a simpler hypothesis, because a simpler hypothesis that fits the data equally well is more likely or plausible than a complex one⁵.

To minimize overfitting, we can either set limits on the trees before building them (pre-pruning), or reduce the tree by removing branches that do not significantly contribute (post-pruning).

NOTES

• In other words, if decision trees are not pruned, they have a high risk of overfitting to the training data⁵.

Dataset Example: X

load_breast_cancer

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 569 entries, 0 to 568
Data columns (total 30 columns):
 #   Column                   Non-Null Count  Dtype  
---  ------                   --------------  -----  
 0   mean radius              569 non-null    float64
 1   mean texture             569 non-null    float64
 2   mean perimeter           569 non-null    float64
 3   mean area                569 non-null    float64
 4   mean smoothness          569 non-null    float64
 5   mean compactness         569 non-null    float64
 6   mean concavity           569 non-null    float64
 7   mean concave points      569 non-null    float64
 8   mean symmetry            569 non-null    float64
 9   mean fractal dimension   569 non-null    float64
 10  radius error             569 non-null    float64
 11  texture error            569 non-null    float64
 12  perimeter error          569 non-null    float64
 13  area error               569 non-null    float64
 14  smoothness error         569 non-null    float64
 15  compactness error        569 non-null    float64
 16  concavity error          569 non-null    float64
 17  concave points error     569 non-null    float64
 18  symmetry error           569 non-null    float64
 19  fractal dimension error  569 non-null    float64
 20  worst radius             569 non-null    float64
 21  worst texture            569 non-null    float64
 22  worst perimeter          569 non-null    float64
 23  worst area               569 non-null    float64
 24  worst smoothness         569 non-null    float64
 25  worst compactness        569 non-null    float64
 26  worst concavity          569 non-null    float64
 27  worst concave points     569 non-null    float64
 28  worst symmetry           569 non-null    float64
 29  worst fractal dimension  569 non-null    float64
dtypes: float64(30)
memory usage: 133.5 KB

None

Extra

You can explore the data below although I recomend limiting the number of features to plot for ease.

Pre-Pruning

An a priori limit on nodes, or tree depth, is often set to avoid overfitting due to a deep tree^4,5.

Notes

• Another one is to stop growing if a split is not statistically significant (e.g., $\chi^2$ test)⁵. However this is not yet availble in sklearn, although I'm sure you can find some code for it somewhere on the internet.

TODO

• turn this into a function
• pull these numbers out of the gridsearch and dont do them again as they are not fully correct!

We could also set a minimum number of data points for each node⁵.

Post-Pruning

In general, post-pruning consists of going back through the tree once it has been created and removing branches that do not significantly contribute to the error reduction and replacing them with leaf nodes⁶

Two common approaches are reduced-error pruning and cost-complexity pruning

• Reduced-error pruning⁵
  • Greedily remove nodes based on validation set performance
  • Generally improves performance but can be problematic for limited data set sizes.
• Cost-complexity pruning⁵
  • Recursively finds the node with the “weakest link”.
  • Nodes are characterized by $\alpha \geq 0$, and nodes with the smallest effective $\alpha$ are pruned first⁷.
  • The trees are then defined as $I + \alpha|N|$, where $I$ is an impurity measure, such as the total misclassification rate of the terminal nodes, $\alpha$ is a tuning parameter, and $|N|$ is the total number of nodes⁵.

Cost-complexity pruning⁷

Using Scikit-learn, we can recursively fit a complex tree with no prior pruning and have a look at the effective alphas and the corresponding total leaf impurities at each step of the pruning process.

As alpha increases, more of the tree is pruned, thus creating a decision tree that generalizes better.

We can select the alpha that reduces the distance between the train and validation scores.

Notes

• In Scikit-learn 0.22 the parameter ccp_alpha was introduced (short for Cost Complexity Pruning- Alpha)
• DecisionTreeClassifier.cost_complexity_pruning_path returns the effective alphas and the corresponding total leaf impurities at each step of the pruning process.

Then we can train a decision tree using the chosen effective alpha.

Other Algorithms

ID3 - Iterative Dichotomizer 3⁸

• Splitting Criterion: Maximises Information Gain/ Minimises Entropy
• Number of Splits: Multiway
• Variables: Discrete binary and multi-category features
• Pruning: None

C4.5⁹

• Splitting Criterion: Maximises Information Gain/ Minimises Entropy
• Number of Splits: Multiway
• Variables: Discrete & Continuous (expensive)
• Pruning: Post-Pruning
• can handle missing data
• C5.0 is the latest release version under a proprietary license

Notes

• you can read an introduction to ID3 in pgs.27-30 of Burkov (2019)

Associated Exercises

Now might be a good time to try exercises X-X.

References

1. James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani. An introduction to statistical learning. Vol. 112. New York: springer, 2013.
2. Gorman KB, Williams TD, Fraser WR (2014). Ecological sexual dimorphism and environmental variability within a community of Antarctic penguins (genus Pygoscelis). PLoS ONE 9(3):e90081. https://doi.org/10.1371/journal.pone.0090081
3. Géron, A. (2017). Hands-on machine learning with Scikit-Learn and TensorFlow: concepts, tools, and techniques to build intelligent systems. " O'Reilly Media, Inc.".
4. Raschka, Sebastian, and Vahid Mirjalili. Python Machine Learning, 2nd Ed. Packt Publishing, 2017.
5. https://github.com/rasbt/stat479-machine-learning-fs19/blob/master/06_trees/06-trees__notes.pdf
6. Burkov, A. (2019). The hundred-page machine learning book (Vol. 1). Canada: Andriy Burkov.
7. https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#:~:text=Cost%20complexity%20pruning%20provides%20another,the%20number%20of%20nodes%20pruned.
8. Quinlan, J. R. (1986). Induction of decision trees. Machine learning, 1 (1), 81-106
9. Quinlan, J. R. (1993). C4. 5: Programming for machine learning. Morgan Kauffmann, 38, 48.
10. https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html#sklearn.tree.DecisionTreeClassifier